In the exploration of subterranean reservoirs many reservoir parameters describing for example geology or fluid distributions are not accessible through direct measurement. Instead, these parameters have to be derived simultaneously using typically a plurality of similar measurements combined with mathematical evaluation methods referred to as inversion methods. Given often large number of unknown parameters, the inversion problem generally suffers from being non-unique. Such inversions are known to be ill-posed problem in the Hadamard sense.
Taking for example inverse problems arising from potential field surveys such as gravity, magnetic or electro-magnetic surveys, the ambiguity arises in several ways. The first one is the inherent ambiguity caused by the physics of the problem that permits an infinity of solutions to produce a given potential field data. For example in the case of gravitational surveys, it has been demonstrated that there are infinitely many different distributions of density that reproduce the same set of gravitational field data or measurements.
The second source of ambiguity is the finite number of observed data. In most cases, the amount of data collected is limited and does not contain sufficient information for a unique solution. The inversion problem remains underdetermined.
The third source is the uncertainty of knowledge. In a realistic experiment, the observed data always have experimental uncertainty. Even the physical theories which define the forward modeling, i.e. the generation of a data set from a given set of parameters or model, are often an approximation or idealization of the reality.
The non-uniqueness can be handled by adding other independent information either mathematically based such as regularization functions or geologically- and geophysically-based constraints. Several methods have been developed to better handle inversion problems. These known methods can be broadly split into two main categories: iterative and probabilistic. During the inversion process, typically only one of these two approaches is systematically used.
The different methods that were introduced to add enough a priori information to restrict the solutions within a region of the parameter space that is considered physically reasonable are distinguished by the type of information added to solve the inversion problem. In the first approach the formation is divided into a large number of cells of a finite size. The algorithm produces a solution by minimizing an objective function of the model subject to fitting the data. The model objective function introduces suitable constraints and regularization operators needed to reduce the undesirable models and to control the geometric shape of the solution.
The many published examples of such an approach to solving the inverse problem include Y. Li and Oldenburg, D. W., 3D inversion of magnetic data, Geophysics, 61 (1996), 2, 394-408, who introduce an objective function which incorporates a reference model such that the constructed model keeps close to the reference model. It imposes smoothness in the three spatial directions, and it has a depth weighting designed to distribute the sensitivity according to the distance from the receivers. Others examples like W. R. Green, Inversion of gravity profiles by use of a Backus-Gilbert approach. Geophysics, 40 (1975), 763-772 minimize a weighted model norm with respect to a reference model. This allows guiding the inversion by varying the weighting according to the available information. And B. J. Last and Kubik. K., Compact gravity inversion: Geophysics, 48 (1983), 713-721 minimize the total volume of the causative body in order to maximize the compactness of the estimated model.
The second approach to restrict the solutions within a region of the parameter space attempts to make use of all the available geological, geophysical or logs data in order to penalize models that do not match all given data. The inversion algorithm produces a model that satisfies simultaneously the available information. For example U.S. Pat. No. 6,502,037(B1) to Jorgensen et al introduces techniques to correct the density model using density logs and seismic data. The density logs serve essentially to select better initial guess of density model. And W. Hu, Abubakar, A. and Habashy, T., Joint inversion algorithm for electromagnetic and seismic data, SEG Expanded Abstracts 26, 1745, San Antonio (2007) present an inversion strategy for reservoir properties identification by using simultaneously cross-well EM and cross-well seismic observation. D. Colombo and De Stefano, M, Geophysical modeling via simultaneous joint inversion of seismic, gravity, and electromagnetic data: application to prestack depth imaging, The Leading Edge (2007), 26, 326-331 use simultaneously the seismic and gravity or EM measurements to enhance the resolution of the structural inversion results.
The methods as described so far can be broadly categorized as iterative approaches. The second group of methods for solving inverse problems are often referred to as probabilistic or Bayesian approaches. Again there are numerous publications which present methods of the probabilistic or Bayesian approach. Details can be taken for example from the tutorial of K. Mosegaard and A. Tarantola, Probabilistic Approach to Inverse Problems, In: International Handbook of Earthquake and Engineering Seismology Volume 81A (2002) Appendix B9, 237-265, which covers solutions of the inversion problem for potential field data.
However in many cases the probabilistic approaches share the general weaknesses of ill-posed problems as described above. The information added in the inversion is often not sufficient to stabilize the solution and therefore joint inversion approaches are also not necessarily capable of producing a unique and acceptable solution.
In view of the known art, it is seen as an object of the invention to provide methods of solving the inverse problem, particularly for potential field related measurements, and thus improving the accuracy or reducing the uncertainty of the parameters derived from the solution.